Lagrange constrained model optimization

Once the objective and the constraints have been set, a mathematical model of the process can be subjected to a search strategy to find the optimum. Simple calculus is adequate for some problems, or Lagrange multipliers can be used for constrained extrema. When a Rill plant simulation can be made, various alternatives can be put through the computer. Such an operation is called jlowsheeting. A chapter is devoted to this topic by Edgar and Himmelblau Optimization of Chemical Processes, McGraw-HiU, 1988) where they list a number of commercially available software packages for this purpose, one of the first of which was Flowtran. 

If the model is a linear plus pair-wise interactions model, the solution can easily be shown to be one of the vertices of the hypercube in the hypercube approach. If the model is a quadratic one, and the optimum (according to the model) is not inside the hypjercube, the solution is a point on one of the edges of the hypercube and a point on the hyp>ersphere in the hypersphere approach. In both approaches, the solution is found most easily using some iterative constrained optimization tool, e.g. Excel s Solver Tool. In the latter (hypersphere) approach, it is easy to show, using the Lagrange multiplier technique of constrained... 

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